If A its nxn Matrix and A^4 - A^3 + A^2 - A + Ιnxn = 0nxn, prove Α^-1 = -Α^4

Thanks!

Dimitristhym
Oct 13, 2018

#1**+4 **

"*If A its nxn Matrix and A^4 - A^3 + A^2 - A + Ιnxn = 0nxn, prove Α^-1 = -Α^4*"

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Alan
Oct 13, 2018

#5**+1 **

You know the question better than me, are you supposed to assume that A is invertible? It IS possible to prove that A is invertible, so I don't think you're supposed to assume that.

Guest Oct 13, 2018

#6**+1 **

First you must prove that,A is invertible because if it don't we can't prove Α^-1 = -Α^4

Dimitristhym
Oct 13, 2018

#8**+1 **

I think i know how to prove it im in 2nd year in Greek university in Math science but this is a lesson for 1st year

and I'm doing it again!

Dimitristhym
Oct 13, 2018

#9**+1 **

Its -A^4 + A^3 - A^2 + A = I <=>

<=> A( -A^3 + A^4 - A + I) =I

so ( -A^3 + A^4 - A + I)=A^-1

Dimitristhym
Oct 13, 2018

#10**+1 **

Yes I think you got it, But just to be sure:

You got the equation A*(-A^{3}+A^{2}-A^{1}+I)=I. We can replace -A^{3}+A^{2}-A^{1}+I with "B". So we got A*B=I. I is an invertible matrix, and if a multiplication of two matrices results in an invertible matrix this means that both matrices are also invertible, meaning that A is invertible.

Guest Oct 13, 2018

#11**+1 **

Yes exactly from definition of invertible matrix!

Thanks guys for your tips!

Dimitristhym
Oct 13, 2018